Can a CNAME point to an NS record

I'd first make sure you're getting ns1/2.provider-dns.com back as an answer to this query:

dig subdomain.example.com ns

You might also want to check that the zone is loaded on the nameservers to which you've delegated. Check that you get a correct serial number back:

dig @ns1.provider-dns.com subdomain.example.com soa

If that isn't working, check with whomever is running ns1/2.provider-dns.com to make sure your zone is being loaded.

DNS data is cached in servers around the world. Checking your servers for changes is likely to mislead you as to whether a change has been made or not.

Looking at your data from here shows that your changes have been made. It looks like your TTL (Time To Live) is about four hours so your changes should be visible to you as well.

When making major changes it is best to make changes in a manner that uses the old and new addresses for a transitional period. One approach would be to publish both the old and new ip addresses at least on TTL period. Then when the transition occurs DNS will have the new addresses to use. Once the transition is done the old IP addresses can be removed.

It is common to change the TTL to a low value prior to the change so that cached data will be refreshed quickly. Scaling down the TTL over a period of time can be used limit the period when most requests are handled by your listed name servers.

EDIT: You can check what various name servers have in their cache with the nslookup or host (newer Linux/Unix distributions) command. You could try google (8.8.4.4 or 8.8.8.8) or OpenDNS (208.67.220.220 or 208.67.222.222) to see what some major caches have. The command for one of the name servers listed above is:

nslookup  ns1.vertigo.bm 8.8.8.8

The timetable I use for scaling down TTL is to reduce it by half whenever the time until the change is twice the current TTL. Keep the TTL at least five or ten minutes. Add the new addresses to the active nameservers 10 to 20 minutes in advance of the change. Remove the old entries after the change has been accomplished.